What is the effect of adding more independent variables to a regression model
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Many of the predictor variables are statistically significantly associated with birth weight. Male infants are approximately 175 grams heavier than female infants, adjusting for gestational age, mother's age and mother's race/ethnicity. Gestational age is highly significant (p=0.0001), with each additional gestational week associated with an increase of 179.89 grams in birth weight, holding infant gender, mother's age and mother's race/ethnicity constant. Mother's age does not reach statistical significance (p=0.6361). Mother's race is modeled as a set of three dummy or indicator variables. In this analysis, white race is the reference group. Infants born to black mothers have lower birth weight by approximately 140 grams (as compared to infants born to white mothers), adjusting for gestational age, infant gender and mothers age. This difference is marginally significant (p=0.0535). There are no statistically significant differences in birth weight in infants born to Hispanic versus white mothers or to women who identify themselves as other race as compared to white.
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Multiple linear regression (MLR), also known simply as multiple regression, is a statistical technique that uses several explanatory variables to predict the outcome of a response variable. The goal of multiple linear regression is to model the linear relationship between the explanatory (independent) variables and response (dependent) variables. In essence, multiple regression is the extension of ordinary least-squares (OLS) regression because it involves more than one explanatory variable. Key Takeaways
Formula and Calculation of Multiple Linear Regressionyi=β0+β1xi1+β2xi2+...+βpxip+ϵwhere, for i=n observations:yi=dependent variablexi=explanatory variablesβ0=y-intercept (constant term)βp=slope coefficients for each explanatory variableϵ=the model’s error term (also known as the residuals)\begin{aligned}&y_i = \beta_0 + \beta _1 x_{i1} + \beta _2 x_{i2} + ... + \beta _p x_{ip} + \epsilon\\&\textbf{where, for } i = n \textbf{ observations:}\\&y_i=\text{dependent variable}\\&x_i=\text{explanatory variables}\\&\beta_0=\text{y-intercept (constant term)}\\&\beta_p=\text{slope coefficients for each explanatory variable}\\&\epsilon=\text{the model's error term (also known as the residuals)}\end{aligned}yi=β0+β1xi1+β2xi2+...+βpxip+ϵwhere, for i=n observations:yi=dependent variablexi=explanatory variablesβ0=y-intercept (constant term)βp=slope coefficients for each explanatory variableϵ=the model’s error term (also known as the residuals) What Multiple Linear Regression Can Tell YouSimple linear regression is a function that allows an analyst or statistician to make predictions about one variable based on the information that is known about another variable. Linear regression can only be used when one has two continuous variables—an independent variable and a dependent variable. The independent variable is the parameter that is used to calculate the dependent variable or outcome. A multiple regression model extends to several explanatory variables. The multiple regression model is based on the following assumptions:
The coefficient of determination (R-squared) is a statistical metric that is used to measure how much of the variation in outcome can be explained by the variation in the independent variables. R2 always increases as more predictors are added to the MLR model, even though the predictors may not be related to the outcome variable. R2 by itself can't thus be used to identify which predictors should be included in a model and which should be excluded. R2 can only be between 0 and 1, where 0 indicates that the outcome cannot be predicted by any of the independent variables and 1 indicates that the outcome can be predicted without error from the independent variables. When interpreting the results of multiple regression, beta coefficients are valid while holding all other variables constant ("all else equal"). The output from a multiple regression can be displayed horizontally as an equation, or vertically in table form. Example of How to Use Multiple Linear RegressionAs an example, an analyst may want to know how the movement of the market affects the price of ExxonMobil (XOM). In this case, their linear equation will have the value of the S&P 500 index as the independent variable, or predictor, and the price of XOM as the dependent variable. In reality, multiple factors predict the outcome of an event. The price movement of ExxonMobil, for example, depends on more than just the performance of the overall market. Other predictors such as the price of oil, interest rates, and the price movement of oil futures can affect the price of XOM and stock prices of other oil companies. To understand a relationship in which more than two variables are present, multiple linear regression is used. Multiple linear regression (MLR) is used to determine a mathematical relationship among several random variables. In other terms, MLR examines how multiple independent variables are related to one dependent variable. Once each of the independent factors has been determined to predict the dependent variable, the information on the multiple variables can be used to create an accurate prediction on the level of effect they have on the outcome variable. The model creates a relationship in the form of a straight line (linear) that best approximates all the individual data points. Referring to the MLR equation above, in our example:
The least-squares estimates—B0, B1, B2…Bp—are usually computed by statistical software. As many variables can be included in the regression model in which each independent variable is differentiated with a number—1,2, 3, 4...p. The multiple regression model allows an analyst to predict an outcome based on information provided on multiple explanatory variables. Still, the model is not always perfectly accurate as each data point can differ slightly from the outcome predicted by the model. The residual value, E, which is the difference between the actual outcome and the predicted outcome, is included in the model to account for such slight variations. Assuming we run our XOM price regression model through a statistics computation software, that returns this output:  An analyst would interpret this output to mean if other variables are held constant, the price of XOM will increase by 7.8% if the price of oil in the markets increases by 1%. The model also shows that the price of XOM will decrease by 1.5% following a 1% rise in interest rates. R2 indicates that 86.5% of the variations in the stock price of Exxon Mobil can be explained by changes in the interest rate, oil price, oil futures, and S&P 500 index. The Difference Between Linear and Multiple RegressionOrdinary linear squares (OLS) regression compares the response of a dependent variable given a change in some explanatory variables. However, a dependent variable is rarely explained by only one variable. In this case, an analyst uses multiple regression, which attempts to explain a dependent variable using more than one independent variable. Multiple regressions can be linear and nonlinear. Multiple regressions are based on the assumption that there is a linear relationship between both the dependent and independent variables. It also assumes no major correlation between the independent variables. What Makes a Multiple Regression Multiple?A multiple regression considers the effect of more than one explanatory variable on some outcome of interest. It evaluates the relative effect of these explanatory, or independent, variables on the dependent variable when holding all the other variables in the model constant. Why Would One Use a Multiple Regression Over a Simple OLS Regression?A dependent variable is rarely explained by only one variable. In such cases, an analyst uses multiple regression, which attempts to explain a dependent variable using more than one independent variable. The model, however, assumes that there are no major correlations between the independent variables. Can I Do a Multiple Regression by Hand?It's unlikely as multiple regression models are complex and become even more so when there are more variables included in the model or when the amount of data to analyze grows. To run a multiple regression you will likely need to use specialized statistical software or functions within programs like Excel. What Does It Mean for a Multiple Regression to Be Linear?In multiple linear regression, the model calculates the line of best fit that minimizes the variances of each of the variables included as it relates to the dependent variable. Because it fits a line, it is a linear model. There are also non-linear regression models involving multiple variables, such as logistic regression, quadratic regression, and probit models. How Are Multiple Regression Models Used in Finance?Any econometric model that looks at more than one variable may be a multiple. Factor models compare two or more factors to analyze relationships between variables and the resulting performance. The Fama and French Three-Factor Mod is such a model that expands on the capital asset pricing model (CAPM) by adding size risk and value risk factors to the market risk factor in CAPM (which is itself a regression model). By including these two additional factors, the model adjusts for this outperforming tendency, which is thought to make it a better tool for evaluating manager performance. What happens when we add more variables to a linear regression model?Adding more variables will increase R Squared whether or not the added variables have any statistically significant effect on the dependent variable. On the other hand, adjusted R Squared can increase or decrease.
What happens if you have too many independent variables?Overfitting occurs when too many variables are included in the model and the model appears to fit well to the current data. Because some of variables retained in the model are actually noise variables, the model cannot be validated in future dataset.
What is the purpose of including multiple independent variables in a regression model?The objective of multiple regression analysis is to use the independent variables whose values are known to predict the value of the single dependent value. Each predictor value is weighed, the weights denoting their relative contribution to the overall prediction.
Can you have too many independent variables in a regression?Many difficulties tend to arise when there are more than five independent variables in a multiple regression equation. One of the most frequent is the problem that two or more of the independent variables are highly correlated to one another. This is called multicollinearity.
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